What are marginal fields in CFT?

In this article they call weight $(h,\bar{h})=(1,1)$ fields marginal.

Why are these fields called marginal? Why are they to be distinguished.

Answer

This terminology comes from renormalization group flow, where one has relevant, marginal, and irrelevant operators.

In CFT, operators with conformal weight $(1, 1)$ are known as marginal operators. More generally, operators of conformal weight $(h, \bar{h})$ are said to be relevant if $h + \bar{h} < 2$ and irrelevant if $h + \bar{h} > 2$. A (necessarily marginal) operator that preserves conformality is called truly marginal, or exactly marginal, etc, cf. Ref. 1.

References:

1. P. Ginsparg, Applied Conformal Field Theory, arXiv:hep-th/9108028; Section 8.6.